Linear functional
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In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. Specifically, if V is a vector space over a field k, then a linear functional is a linear function from V to k.
The set of all linear functionals from V to k, HomK(V,k), is itself a k-vector space. This space is called the dual space of V. If V is a topological vector space the space of continuous linear functionals, the continuous dual, is often simply called the dual space. If V is a Banach space then so is its continuous dual.
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation
- <math>f \mapsto \int_a^b f(x)\, dx<math>
is a linear functional from the space of integrable functions to the reals.
Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra-ket notation.